How can I bend a mesh so that it curves around other object?
As you correctly guessed, you can use the simple deform modifier, with the bend option. However, you have to change the direction and origin of the bend You can control this by creating an empty and positioning it close to the button, if you select the empty in the origin property of the modifier. Here, I created two deform modifiers, controlled by two empties:



The two empties were made the children of the button, and rotated so that the bending happened in the right direction (to find the right direction, I had to play with the rotation a bit - there is no easy way to understand the correct orientation):



The modifiers look like this (again, I had to try different angles to find a good value):



Here is the .blend file: